The **Mandelbrot set** is the set of **complex numbers** $c$ for which the function $f_{c}(z) = z^{2} + c$ does **not** diverge when iterated from $z_{0} = 0$. This means that starting with $z = 0$, if the iterations of the function $f_{c}(z) = z^{2} + c$ do not cause $\left\lvert z \right\rvert$ to grow beyond a certain threshold (usually $2$) after many iterations, then the point $c$ is part of the Mandelbrot set.
### 1. Choose a complex constant $c$
> * Fot the Mandelbrot set, $c$ **is the parameter** that changes with each point we check in the complex plane. The set is defined by **which value of $c$ produce bounded iterates**.
### 2. Define the complex plane
### 3. Start with $z_{0} = 0$
### 4. Iterate the function $f_{c}(z) = z^{2} + c$
### 5. Check if $\left\lvert z \right\rvert$ (the modulus of $z$) escapes
### 6. Repeat the iteration for a maximum number of iterations
### 7. Color the point
### 8. Repeat for all points in the complex plane
# Julia set
## What is a Julia set?
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> * **Second iteration**: Now use $z_{1} = ... + ... \imath$ and apply the same function.
> * **Finally**, you repeat this process for several iterations.
### 5. Check if $\left\lvert z \right\rvert$ (the module of $z$) escapes
### 5. Check if $\left\lvert z \right\rvert$ (the modulus of $z$) escapes
> * **Escape condition**: If at any point during the iterations $\left\lvert z_{n} \right\rvert$ (the magnitude of $z_{n}$) exceeds a threshold (commonly $2$), we say the point $z$ escapes and is **not** in the Julia set we stop iterating.
